One thing to remember is that ideologies can be successfully introduced because they appeal to people (by providing them with the proper incentives), and successfully maintained in the long term because they provide or protect something that people value.
Another thing to remember is that economics actually is the study of incentives and value, and doesn’t just apply to finance etc. When deciding wether you value honor over survival, you are making an economic decision. When considering that your belief in a god, or in the supreme importance of you nation, is your ultimate motivation (your highest incentive), you are performing an economic analysis.
An ideology that does not provide incentives and does not, at its core, provide or maintain something we value will not be successful. Incentives are the means, and values are the ends. Even an ideology that uses psychology as a means will still have to accept that psychological incentives are still incentives, and moreover, it will still have to work towards the same ends: things we value.
It seems to me, then, that it is not actually possible to take “economics” out of ideology. We can only make the economics less obvious, and apply them to non-monetary (or even non-material) issues. So yes, our modern reliance on economics when it comes to political discourse may be a historical fluke, but our reliance on the economic evaluations themselves is not. We are always making economic choices, even when we are unaware of it. The very act of choosing between two options is an economic evaluation in itself.
God or country?
Duty or integrity?
Love or honor?
Well; which do you value higher? And what are your incentives? That's what it's all about. Economics cannot be discarded, or even truly disregarded.